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Let $X$ be a curve (proper, smooth, ...) over a finite field $\mathbb F_q$ where $q$. Suppose also that $\mathbb F_q$ contains the $p$-th roots of unity, in this case we have the following (unique) chain of field extensions: $$\mathbb F_q \subset \mathbb F_{q^p} \subset \dots \subset \mathbb F_q^{(p)}$$ where each step in the extension is degree $p$ and the union of all these extensions is defined to be $\mathbb F_q^{(p)}$.

I am interested in the $\ell$ (possibly equal to $p$) torsion of the degree $0$ Picard group of $X_{\mathbb F_q^{(p)}}$. If we consider the algebraic closure $X_{\overline{\mathbb F_q}}$, then the $\ell$ torsion here looks like $P = (\mathbb Q_\ell/\mathbb Z_\ell)^{2\lambda}$ for $\lambda \leq g(X)$, the genus.

One strategy to get information about the Picard group of $X_{\mathbb F_q^{(p)}}$ would be to consider the Galois group $\Gamma = \mathrm{Gal}(\overline{\mathbb F_q}/\mathbb F_q^{(p)}) \cong \prod_{p' \neq \ell}\mathbb Z_{p'}$ and it's action on $P$.

Can this strategy prove the following:

1) If $\ell = p$, then the $\ell$-torsion in the Picard group over $\mathbb F_q^{(p)}$ is of the form $(\mathbb Q_\ell/\mathbb Z_\ell)^r$ for $r \leq 2\lambda$.

2) If $\ell \neq p$, then the $\ell$-torsion in the Picard group over $\mathbb F_q^{(p)}$ is finite.

Ideally, I would like to avoid any knowledge of the zeta function of curves.

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    $\begingroup$ Have you looked at [Rosen, Number Theory in Function Fields], Theorem 11.5: $\mathrm{ord}_\ell |\mathrm{Pic}^0_{C/\mathbf{F}_q}(\mathbf{F}_{q^{\ell^n}})| = \lambda_\ell n + \nu_\ell$ for $n \geq n_0$ and Theorem 11.6: $\mathrm{ord}_{\ell'}(...)$ are bounded and Proposition 11.16. $\endgroup$
    – user19475
    Sep 25, 2018 at 4:37
  • $\begingroup$ By the way, the $\ell$-primary component of $\mathrm{Pic}^0$ over an algebraically closed field is $(\mathbf{Q}_\ell/\mathbf{Z}_\ell)^{2g}$ for $\ell \neq \mathrm{char}$ and $(\mathbf{Q}_p/\mathbf{Z}_p)^{r}$ with $0 \leq r \leq g$ otherwiese. $\endgroup$
    – user19475
    Sep 25, 2018 at 5:54
  • $\begingroup$ What do you mean in the second sentence when you write that your field of characteristic $p$ contains $p^{\text{th}}$ roots of unity? The polynomial $x^p-1$ factors as $(x-1)^p$ in characteristic $p$. $\endgroup$ Sep 25, 2018 at 8:20
  • $\begingroup$ @jasonstarr q is supposed to be a prime power of a prime other than p... Sorry! $\endgroup$
    – Asvin
    Sep 25, 2018 at 14:45
  • $\begingroup$ @tke thanks for the reference. That was indeed the proof I was trying to avoid, in particular I would like to avoid knowledge about the zeta function of the curve but this might be asking too much. $\endgroup$
    – Asvin
    Sep 25, 2018 at 18:03

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